Progress in Mathematics
Volume 255
Series Editors
Hyman Bass
Joseph Oesterl´e
Alan Weinstein
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Representation Theory
and Automorphic Forms
Toshiyuki Kobayashi
Wilfried Schmid
Jae-Hyun Yang
Editors
Birkhăauser
Boston ã Basel ã Berlin
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Toshiyuki Kobayashi
RIMS, Kyoto University
Sakyo-ku, Kyoto, 606-8502
Japan
Wilfried Schmid
Department of Mathematics
Harvard University
Cambridge, MA 02138
U.S.A.
Jae-Hyun Yang
Department of Mathematics
Inha University
Incheon 402-751
Republic of Korea
Mathematics Subject Classifications (2000): 11R39, 11R42, 14G35, 22E45, 22E50, 22E55
Library of Congress Control Number: 2007933203
ISBN-13: 978-0-8176-4505-2
e-ISBN-13: 978-0-8176-4646-2
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c 2008 Birkhăauser Boston
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Irreducibility and Cuspidality
Dinakar Ramakrishnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The first step in the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The second step in the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Galois representations attached to regular, selfdual cusp forms on GL(4)
5 Two useful lemmas on cusp forms on GL(4) . . . . . . . . . . . . . . . . . . . . . .
6 Finale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
5
15
16
18
20
21
25
2 On Liftings of Holomorphic Modular Forms
Tamotsu Ikeda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Fourier coefficients of the Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . .
3 Kohnen plus space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Lifting of cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Relation to the Saito–Kurokawa lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Hermitian modular forms and hermitian Eisensetein series . . . . . . . . . . .
8 The case m = 2n + 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 The case m = 2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 The case m = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
30
32
33
34
35
37
39
40
40
41
42
3 Multiplicity-free Theorems of the Restrictions of Unitary Highest
Weight Modules with respect to Reductive Symmetric Pairs
Toshiyuki Kobayashi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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vi
Contents
1 Introduction and statement of main results . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Main machinery from complex geometry . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Proof of Theorem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Uniformly bounded multiplicities — Proof of Theorems B and D . . . . . 70
6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7 Finite-dimensional cases — Proof of Theorems E and F . . . . . . . . . . . . . 83
8 Generalization of the Hua–Kostant–Schmid formula . . . . . . . . . . . . . . . . 89
9 Appendix: Associated bundles on Hermitian symmetric spaces . . . . . . . 103
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 The Rankin–Selberg Method for Automorphic Distributions
Stephen D. Miller and Wilfried Schmid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Standard L-functions for S L(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Pairings of automorphic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 The Rankin–Selberg L-function for G L(2) . . . . . . . . . . . . . . . . . . . . . . . .
5 Exterior Square on G L(4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
111
115
121
128
137
149
5 Langlands Functoriality Conjecture and Number Theory
Freydoon Shahidi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Modular forms, Galois representations and Artin L-functions . . . . . . . .
3 Lattice point problems and the Selberg conjecture . . . . . . . . . . . . . . . . . .
4 Ramanujan conjecture for Maass forms . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Sato–Tate conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Functoriality for symmetric powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Functoriality for classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Ramanujan conjecture for classical groups . . . . . . . . . . . . . . . . . . . . . . . .
9 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
151
152
156
158
159
161
163
164
166
169
6 Discriminant of Certain K 3 Surfaces
Ken-Ichi Yoshikawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction – Discriminant of elliptic curves . . . . . . . . . . . . . . . . . . . . . .
2 K 3 surfaces with involution and their moduli spaces . . . . . . . . . . . . . . . .
3 Automorphic forms on the moduli space . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Equivariant analytic torsion and 2-elementary K 3 surfaces . . . . . . . . . . .
5 The Borcherds products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Borcherds products for odd unimodular lattices . . . . . . . . . . . . . . . . . . . .
7 K 3 surfaces of Matsumoto–Sasaki–Yoshida . . . . . . . . . . . . . . . . . . . . . . .
8 Discriminant of quartic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
175
178
180
182
184
186
188
200
209
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Preface
Over the last half century, deep connections between representation theory and automorphic forms have been established, using a wide range of methods from algebra,
geometry and analyis. In light of these developments, Changho Keem, Toshiyuki
Kobayashi and Jae-Hyun Yang organized an international symposium entitled “Representation Theory and Automorphic Forms”, with the hope that a broad discussion
of recent ideas and techniques would lead to new breakthroughs in the field. The
symposium was held at Seoul National University, Republic of Korea, February 14–
17, 2005.
This volume is an outgrowth of the symposium. The lectures cover a variety of
aspects of representation theory and autmorphic forms, among them, a lifting of elliptic cusp forms to Siegel and Hermitian modular forms (T. Ikeda), systematic and
synthetic applications of the original theory of “visible actions” on complex manifolds to “multiplicity-free” theorems, in particular, to branching problems for reductive symmetric pairs (T. Kobayashi), an adaption of the Rankin–Selberg method to
the setting of automorphic distributions (S. Miller and W. Schmid), recent developments in the Langlands functoriality conjecture and their relevance to certain conjectures in number theory, such as the Ramanujan and Selberg conjectures (F. Shahidi),
cuspidality-irreducibility relation for automorphic representations (D. Ramakrishnan), and applications of Borcherds automorphic forms to the study of discriminants
of certain K 3 surfaces with involution that arise from the theory of hypergeometric functions (K.-I. Yoshikawa). By presenting some of the most active topics in the
field, the editors hope that this volume will serve as an up-to-date introduction to the
subject.
Acknowledgments
We thank the invited speakers for their enthusiastic lectures and the articles they have
contributed. The referees also deserve our gratitude for their important role.
The symposium was initiated by, and received funding from, the Brain Korea 21
Mathematical Sciences Division of Seoul National University, BK21-MSD-SNU for
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viii
Preface
short. Our thanks go to Hyuk Kim, Director of BK21-MSD-SNU, and to ChongKyu Han, Chairman of Department of Mathematics at SNU, for their support. We
are especially indebted to Sung-Hoon Park, President of the JEI Corporation, and to
Jee Hoon Park, President of JEI Distribution Co., LTD – and coincidentally a friend
of the last-named editor; without the very generous financial contribution of the JEI
Corporation the symposium would have been far more modest in scale.
Jaeyeon Joo and Eun-Soon Hong, secretaries of BK21-MSN-SNU, and DongSoo Shin did a splendid job preparing the symposium and catering to the needs of the
participants. We are grateful also to Ann Kostant and Avanti Paranjpye of Birkhăauser
Boston for their work in publishing this volume.
January 2007
Toshiyuki Kobayashi
Wilfried Schmid
Jae-Hyun Yang
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1
Irreducibility and Cuspidality
Dinakar Ramakrishnan
253-37 Caltech
Pasadena, CA 91125, USA
Summary. Suppose ρ is an n-dimensional representation of the absolute Galois group of Q
which is associated, via an identity of L-functions, with an automorphic representation π of
GL(n) of the adele ring of Q. It is expected that π is cuspidal if and only if ρ is irreducible,
though nothing much is known in either direction in dimensions > 2. The object of this article
is to show for n < 6 that the cuspidality of a regular algebraic π is implied by the irreducibility
of ρ. For n < 5, it suffices to assume that π is semi-regular.
Key words: irreducibility, Galois representations, cuspidality, automorphic representations, general linear group, symplectic group, regular algebraic representations
Subject Classifications: 11F70; 11F80; 22E55
Introduction
Irreducible representations are the building blocks of general, semisimple Galois
representations ρ, and cuspidal representations are the building blocks of automorphic forms π of the general linear group. It is expected that when an object of the
former type is associated to one of the latter type, usually in terms of an identity of
L-functions, the irreducibility of the former should imply the cuspidality of the latter,
and vice versa. It is not a simple matter to prove this expectation, and nothing much is
known in dimensions > 2. We will start from the beginning and explain the problem
below, and indicate a result (in one direction) at the end of the introduction, which
summarizes what one can do at this point. The remainder of the paper will be devoted
to showing how to deduce this result by a synthesis of known theorems and some new
ideas. We will be concerned here only with the so-called easier direction of showing
the cuspidality of π given the irreducibility of ρ, and refer to [Ra5] for a more difficult result going the other way, which uses crystalline representations as well as a
∗ Partially supported by the NSF through the grant DMS-0402044.
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2
Dinakar Ramakrishnan
refinement of certain deep modularity results of Taylor, Skinner–Wiles, et al. Needless to say, easier does not mean easy, and the significance of the problem stems
from the fact that it does arise (in this direction) naturally. For example, π could
be a functorial, automorphic image r (η), for η a cuspidal automorphic representation of a product of smaller general linear groups: H (A) =
j G L(m j , A), with
an associated Galois representation σ such that ρ = r (σ ) is irreducible. If the automorphy of π has been established by using a flexible converse theorem ([CoPS1]),
then the cuspidality of π is not automatic. In [RaS], we had to deal with this question for cohomological forms π on GL(6), with H = GL(2) × GL(3) and r the
Kronecker product, where π is automorphic by [KSh1]. Besides, the main result
(Theorem A below) of this paper implies, as a consequence, the cuspidality of π =
sym4 (η) for η defined by any non-CM holomorphic newform ϕ of weight ≥ 2 relative to 0 (N) ⊂ SL(2, Z), without appealing to the criterion of [KSh2]; here the
automorphy of π is known by [K] and the irreducibility of ρ by [Ri].
Write Q for the field of all algebraic numbers in C, which is an infinite, mysterious Galois extension of Q. One could say that the central problem in algebraic
number theory is to understand this extension. Class field theory, one of the towering achievements of the twentieth century, helps us understand the abelian part of
this extension, though there are still some delicate, open problems even in that well
traversed situation.
Let GQ denote the absolute Galois group of Q, meaning Gal(Q/Q). It is a profinite group, being the projective limit of finite groups Gal(K /Q), as K runs over
number fields which are normal over Q. For fixed K , the Tchebotarev density theorem asserts that every conjugacy class C in Gal(K /Q) is the Frobenius class for
an infinite number of primes p which are unramified in K . This shows the importance of studying the representations of Galois groups, which are intimately tied
up with conjugacy classes. Clearly, every C-representation, i.e., a homomorphism
into GL(n, C) for some n, of Gal(K /Q) pulls back, via the canonical surjection
GQ → Gal(K /Q), to a representation of GQ , which is continuous for the profinite
topology.
Conversely, one can show that every continuous C-representation ρ of GQ is
such a pullback, for a suitable finite Galois extension K /Q. E. Artin associated
an L-function, denoted L(s, ρ), to any such ρ, such that the arrow ρ → L(s, ρ)
is additive and inductive. He conjectured that for any non-trivial, irreducible, continuous C-representation ρ of GQ , L(s, ρ) is entire, and this conjecture is open
in general. Again, one understands well the abelian situation, i.e., when ρ is a 1dimensional representation; the kernel of such a ρ defines an abelian extension of
Q. By class field theory, such a ρ is associated to a character ξ of finite order of
the idele class group A∗ /Q∗ ; here, being associated means they have the same Lfunction, with L(s, ξ ) being the one introduced by Hecke, albeit in a different language. As usual, we are denoting by A = R × A f the topological ring of adeles, with A f = Z ⊗ Q, and by A∗ its multiplicative group of ideles, which can
be given the structure of a locally compact abelian topological group with discrete
subgroup Q∗ .
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1 Irreducibility and Cuspidality
3
Now fix a prime number , and an algebraic closure Q of the field of -adic
numbers Q , equipped with an embedding Q → Q . Consider the set R (n, Q) of
continuous, semisimple representations
ρ : GQ → GL(n, Q ),
up to equivalence. The image of GQ in such a representation is usually not finite, and
the simplest example of that is given by the -adic cyclotomic character χ given by
the action of GQ on all the -power roots of unity in Q. Another example is given
by the 2-dimensional -adic representation on all the -power division points of an
elliptic curve E over Q.
The correct extension to the non-abelian case of the idele class character, which
appears in class field theory, is the notion of an irreducible automorphic representation π of GL(n). Such a π is in particular a representation of the locally compact
group GL(n, A F ), which is a restricted direct product of the local groups GL(n, Qv ),
where v runs over all the primes p and ∞ (with Q∞ = R). There is a corresponding factorization of π as a tensor product ⊗v πv , with all but a finite number of π p
being unramified, i.e., admitting a vector fixed by the maximal compact subgroup
K v . At the archimedean place ∞, π∞ corresponds to an n-dimensional, semisimple
representation σ (π∞ ) of the real Weil group WR , which is a non-trivial extension
of Gal(C/R) by C∗ . Globally, by Schur’s lemma, the center Z (A)
A∗ acts by a
∗
quasi-character ω, which must be trivial on Q by the automorphy of π, and so defines an idele class character. Let us restrict to the central case when π is essentially
unitary. Then there is a (unique) real number t such that the twisted representation
πu := π(t) = π ⊗|·|t is unitary (with unitary central character ωu ). We are, by abuse
of notation, writing | · |t to denote the quasi-character | · |t ◦ det of GL(n, A), where
| · | signifies the adelic absolute value, which is trivial on Q∗ by the Artin product
formula.
Roughly speaking, to say that π is automorphic means πu appears (in a weak
sense) in L 2 (Z (A)GL(n, Q)\GL(n, A), ωu ), on which GL(n, A F ) acts by right
translations. A function ϕ in this L 2 -space whose averages over all the horocycles are zero is called a cusp form, and π is called cuspidal if πu is generated by
the right GL(n, A F )-translates of such a ϕ. Among the automorphic representations
of GL(n, A) are certain distinguished ones called isobaric automorphic representations. Any isobaric π is of the form π1 π2 · · · πr , where each π j is a cuspidal
representation of GL(n j , A), such that (n 1 , n 2 , . . . , nr ) is a partition of n, where
denotes the Langlands sum (coming from his theory of Eisenstein series); moreover,
every constituent π j is unique up to isomorphism. Let A(n, Q) denote the set of isobaric automorphic representations of GL(n, A) up to equivalence. Every isobaric π
has an associated L-function L(s, π) = v L(s, πv ), which admits a meromorphic
continuation and a functional equation. Concretely, one associates at every prime
p where π is unramified, a conjugacy class A(π) in GL(n, C), or equivalently, an
unordered n-tuple (α1, p , α2, p , . . . , αn, p ) of complex numbers so that
n
L(s, π p ) =
(1 − α j, p p−s )−1 .
j =1
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Dinakar Ramakrishnan
If π is cuspidal and non-trivial, L(s, π) is entire; so is the incomplete one L S (s, π)
for any finite set S of places of Q.
Now suppose ρ is an n-dimensional, semisimple -adic representation of GQ =
Gal(Q/Q) corresponds to an automorphic representation π of GL(n, A). We will
take this to mean that there is a finite set S of places including , ∞ and all the
primes where ρ or π is ramified, such that we have
L(s, π p ) = L p (s, ρ ),
∀p ∈
/ S,
(0.1)
where the Galois Euler factor on the right is given by the characteristic polynomial
of Fr p , the Frobenius at p, acting on ρ . When (0.1) holds (for a suitable S), we will
write
ρ ↔ π.
A natural question in such a situation is to ask if π is cuspidal when ρ is irreducible, and vice versa. It is certainly what is predicted by the general philosophy.
However, proving it is another matter altogether, and positive evidence is scarce beyond n = 2.
One can answer this question in the affirmative, for any n, if one restricts to those
ρ which have finite image. In this case, it also defines a continuous, C-representation
ρ, the kind studied by E. Artin ([A]). Indeed, the hypothesis implies the identity of
L-functions
(0.2)
L S (s, ρ ⊗ ρ ∨ ) = L S (s, π × π ∨ ),
where the superscript S signifies the removal of the Euler factors at places in S, and
ρ ∨ (resp. π ∨ ) denotes the contragredient of ρ (resp. π). The L-function on the right
is the Rankin–Selberg L-function, whose mirific properties have been established in
the independent and complementary works of Jacquet, Piatetski-Shapiro and Shalika
([JPSS], and of Shahidi ([Sh1, Sh2]); see also [MW]. A theorem of Jacquet and
Shalika ([JS1]) asserts that the order of pole at s = 1 of L S (s, π × π ∨ ) is 1 iff π
is cuspidal. On the other hand, for any finite-dimensional C-representation τ of GQ ,
one has
(0.3)
−ords=1 L S (s, τ ) = dimC HomGQ (1, τ ),
where 1 denotes the trivial representation of GQ . Applying this with τ = ρ ⊗ ρ ∨
End(ρ), we see that the order of pole of L S (s, ρ ⊗ ρ ∨ ) at s = 1 is 1 iff the only
operators in End(ρ) which commute with the GQ -action are scalars, which means by
Schur that ρ is irreducible. Thus, in the Artin case, π is cuspidal iff ρ is irreducible.
For general -adic representations ρ of GQ , the order of pole at the right edge
is not well understood. When ρ comes from arithmetic geometry, i.e., when it is a
Tate twist of a piece of the cohomology of a smooth projective variety over Q which
is cut out by algebraic projectors, an important conjecture of Tate asserts an analogue
of (0.3) for τ = ρ ⊗ ρ ∨ , but this is unknown except in a few families of examples,
such as those coming from the theory of modular curves, Hilbert modular surfaces
and Picard modular surfaces. So one has to find a different way to approach the
problem, which works at least in low dimensions.
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1 Irreducibility and Cuspidality
5
The main result of this paper is the following:
Theorem A. Let n ≤ 5 and let be a prime. Suppose ρ ↔ π, for an isobaric,
algebraic automorphic representation π of GL(n, A), and a continuous, -adic representation ρ of GQ . Assume
(i) ρ is irreducible
(ii) π is odd if n ≥ 3
(iii) π is semi-regular if n = 4, and regular if n = 5
Then π is cuspidal.
Some words of explanation are called for at this point. An isobaric automorphic representation π is said to be algebraic ([C 1]) if the restriction of σ∞ :=
n
∗
σ (π∞ ( 1−n
2 )) to C is of the form ⊕ j =1 χ j , with each χ j algebraic, i.e., of the form
qj
p
j
z → z z with p j , q j ∈ Z. (We do not assume that our automorphic representations are unitary, and the arrow π∞ → σ∞ is normalized arithmetically.) For n = 1,
an algebraic π is an idele class character of type A0 in the sense of Weil. One says
that π is regular iff σ∞ |C∗ is a direct sum of characters χ j , each occurring with multiplicity one. And π is semi-regular ([BHR]) if each χ j occurs with multiplicity at most
two. Suppose ξ is a 1-dimensional representation of WR . Then, since WRab R∗ , ξ is
defined by a character of R∗ of the form x → |x|w · sgn(x)a(ξ ), with a(ξ ) ∈ {0, 1};
here sgn denotes the sign character of R∗ . For every w, let σ∞ [ξ ] := σ (π∞ ( 1−n
2 ))[ξ ]
denote the isotypic component of ξ , which has dimension at most 2 (resp. 1) if π is
semi-regular (resp. regular), and is acted on by R∗ /R∗+
{±1}. We will call a
semi-regular π odd if for every character ξ of WR , the eigenvalues of R∗ /R∗+ on the
ξ -isotypic component are distinct. Clearly, any regular π is odd under this definition.
See Section 1 for a definition of this concept for any algebraic π, not necessarily
semi-regular.
I want to thank the organizers, Jae-Hyun-Yang in particular, and the staff, of
the International Symposium on Representation Theory and Automorphic Forms in
Seoul, Korea, first for inviting me to speak there (during February 14–17, 2005),
and then for their hospitality while I was there. The talk I gave at the conference
was on a different topic, however, and dealt with my ongoing work with Dipendra
Prasad on selfdual representations. I would also like to thank F. Shahidi for helpful
conversations and the referee for his comments on an earlier version, which led to
an improvement of the presentation. It is perhaps apt to end this introduction at this
point by acknowledging support from the National Science Foundation via the grant
DMS − 0402044.
1 Preliminaries
1.1 Galois representations
For any field k with algebraic closure k, denote by Gk the absolute Galois group of k
over k. It is a projective limit of the automorphism groups of finite Galois extensions
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6
Dinakar Ramakrishnan
E/k. We furnish Gk as usual with the profinite topology, which makes it a compact,
totally disconnected topological group. When k = F p , there is for every n a unique
extension of degree n, which is Galois, and GF p is isomorphic to Z
limn Z/n,
topologically generated by the Frobenius automorphism x → x p .
At each prime p, let G p denote the local Galois group Gal(Q p /Q p ) with inertia
subgroup I p , which fits into the following exact sequence:
1 → I p → G p → GF p → 1.
(1.1.1)
The fixed field of Q p under I p is the maximal unramified extension Qur
p of Q p , which
is generated by all the roots of unity of order prime to p. One gets a natural isomorphism of Gal(Qur
p /Q p ) with GF p . If K /Q is unramified at p, then one can lift the
Frobenius element to a conjugacy class ϕ p in Gal(K /Q).
All the Galois representations considered here will be continuous and finitedimensional. Typically, we will fix a prime , and algebraic closure Q of the field
Q of -adic numbers, and consider a continuous homomorphism
ρ : GQ → GL(V ),
(1.1.2)
where V is an n-dimensional vector space over Q . We will be interested only in
those ρ that are unramified outside a finite set S of primes. Then ρ factors through
a representation of the quotient group G S := G(Q S /Q), where Q S is the maximal
extension of Q which is unramified outside S. One has the Frobenius classes φ p in
/ S, and this allows one to define the L-factors (with s ∈ C)
G S for all p ∈
L p (s, ρ ) = det(I − ϕ p p−s |V )−1 .
(1.1.3)
Clearly, it is the reciprocal of a polynomial in p−s of degree n, with constant term 1,
and it depends only on the equivalence class of ρ . One sets
L S (s, ρ ) =
L p (s, ρ ).
(1.1.4)
p ∈S
/
When ρ is the trivial representation, it is unramified everywhere, and L S (s, ρ ) is
none other than the Riemann zeta function. To define the bad factors at p in S − { },
I
one replaces V in (1.1.3) the subspace V p of inertial invariants, on which ϕ p acts.
We are primarily interested in semisimple representations in this article, which
are direct sums of simple (or irreducible) representations. Given any representation
ρ of GQ , there is an associated semisimplification, denoted ρ ss , which is a direct sum
of the simple Jordan–Holder components of ρ . A theorem of Tchebotarev asserts
the density of the Frobenius classes in the Galois group, and since the local p-factors
of L(s, ρ ) are defined in terms of the inverse roots of ϕ p , one gets the following
standard, but useful result.
Proposition 1.1.5. Let ρ , ρ be continuous, n-dimensional -adic representations
of GQ . Then
ss
L S (s, ρ ) = L S (s, ρ ) ⇒ ρ ss ρ .
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1 Irreducibility and Cuspidality
7
The Galois representations ρ which have finite image are special, and one can
view them as continuous C-representations ρ. Artin studied these in depth and
showed, using the results of Brauer and Hecke, that the corresponding L-functions
admit meromorphic continuation and a functional equation of the form
L ∗ (s, ρ) = ε(s, ρ)L ∗ (1 − s, ρ ∨ ),
(1.1.6)
where ρ ∨ denotes the contragredient representation on the dual vector space, where
L ∗ (s, ρ) = L(s, ρ)L ∞ (s, ρ),
(1.1.7)
with the archimedean factor L ∞ (s, ρ) being a suitable product (shifted) gamma
functions. Moreover,
(1.1.8)
ε(s, ρ) = W (ρ)N(ρ)s−1/2 ,
which is an entire function of s, with the (non-zero) W (ρ) being called the root
number of ρ. The scalar N(ρ) is an integer, called the Artin conductor of ρ, and
the finite set S which intervenes is the set of primes dividing N(ρ). The functional
equation shows that W (ρ)W (ρ ∨ ) = 1, and so W (ρ) = ±1 when ρ is selfdual (which
means ρ ρ ∨ ). Here is a useful fact:
Proposition 1.1.9 ([T]). Let τ be a continuous, finite-dimensional C-representation
of GQ , unramified outside S. Then we have
−ords=1 L S (s, τ ) = HomGQ (1, τ ).
Corollary 1.1.10. Let ρ be a continuous, finite-dimensional C-representation of GQ ,
unramified outside S. Then ρ is irreducible if and only if the incomplete L-function
L S (s, ρ ⊗ ρ ∨ ) has a simple pole at s = 1.
Indeed, if we set
τ := ρ ⊗ ρ ∨
End(ρ),
(1.1.11)
then Proposition 1.1.9 says that the order of pole of L(s, ρ ⊗ ρ ∨ ) at s = 1 is the multiplicity of the trivial representation in End(ρ) is 1, i.e., iff the commutant EndGQ (ρ)
is one-dimensional (over C), which in turn is equivalent, by Schur’s lemma, to ρ
being irreducible. Hence the corollary.
For general -adic representations ρ , there is no known analogue of Proposition 1.1.9, though it is predicted to hold (at the right edge of absolute convergence)
by a conjecture of Tate when ρ comes from arithmetic geometry (see [Ra4], Section 1, for example). Tate’s conjecture is only known in certain special situations,
such as for C M abelian varieties. For the L-functions in Tate’s set-up, say of motivic weight 2m, one does not even know that they make sense at the Tate point
s = m + 1, let alone know its order of pole there. Things get even harder if ρ does
not arise from a geometric situation. One cannot work in too general a setting, and
at a minimum, one needs to require ρ to have some good properties, such as being
unramified outside a finite set S of primes. Fontaine and Mazur conjecture ([FoM])
that ρ is geometric if it has this property (of being unramified outside a finite S) and
is in addition potentially semistable.
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Dinakar Ramakrishnan
1.2 Automorphic representations
Let F be a number field with adele ring A F = F∞ × A F, f , equipped with the
adelic absolute value | · | = | · |A . For every algebraic group G over F, let
G(A F ) = G(F∞ ) × G(A F, f ) denote the restricted direct product v G(Fv ), endowed with the usual locally compact topology. Then G(F) embeds in G(A F ) as a
discrete subgroup, and if Z n denotes the center of GL(n), the homogeneous space
GL(n, F)Z n (A F )\GL(n, A F ) has finite volume relative to the relatively invariant
quotient measure induced by a Haar measure on GL(n, A F ). An irreducible representation π of GL(n, A F ) is admissible if it admits a factorization as a restricted
tensor product ⊗v πv , where each πv is admissible and for almost all finite places
v, πv is unramified, i.e., has a no-zero vector fixed by K v = GL(n, Ov ). (Here, as
usual, Ov denotes the ring of integers of the local completion Fv of F at v.)
Fixing a unitary idele class character ω, which can be viewed as a character of
Z n (A F ), we may consider the space
L 2 (n, ω) := L 2 (GL(n, F)Z n (A F )\GL(n, A F ), ω),
(1.2.1)
which consists of (classes of) functions on GL(n, A F ) that are left-invariant under
GL(n, F), transform under Z n (A F ) according to ω, and are square-integrable modulo GL(n, F)Z (A F ). Clearly, L 2 (n, ω) is a unitary representation of GL(n, A F )
under the right translation action on functions. The space of cusp forms, denoted
L 20 (n, ω), consists of functions ϕ in L 2 (n, ω) which satisfy the following for every
unipotent radical U of a standard parabolic subgroup P = MU :
U (F )\U (A F )
ϕ(ux) = 0.
(1.2.2)
To say that P is a standard parabolic means that it contains the Borel subgroup of
upper triangular matrices in GL(n). A basic fact asserts that L 20 (n, ω) is a subspace
of the discrete spectrum of L 2 (n, ω).
By a unitary cuspidal (automorphic) representation π of GLn (A F ), we will
mean an irreducible, unitary representation occurring in L 20 (n, ω). We will, by abuse
of notation, also denote the underlying admissible representation by π. (To be precise, the unitary representation is on the Hilbert space completion of the admissible
space.) Roughly speaking, unitary automorphic representations of GL(n, A F ) are
those which appear weakly in L 2 (n, ω) for some ω. We will refrain from recalling
the definition precisely, because we will work totally with the subclass of isobaric
automorphic representations, for which one can take Theorem 1.2.10 (of Langlands)
below as their definition.
If π is an admissible representation of GL(n, A F ), then for any z ∈ C, we define
the analytic Tate twist of π by z to be
π(z) := π ⊗ | · |z ,
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(1.2.3)
1 Irreducibility and Cuspidality
9
where | · |z denotes the 1-dimensional representation of GL(n, A F ) given by
g → |det(g)|z = e z log(|det(g)|).
Since the adelic absolute value |·| takes det(g) to a positive real number, its logarithm
is well defined.
In general, by a cuspidal automorphic representation, we will mean an irreducible admissible representation of GL(n, A F ) for which there exists a real number
w, which we will call the weight of π, such that the Tate twist
πu := π(w/2)
(1.2.4)
is a unitary cuspidal representation. Note that the central character of π and of its
unitary avatar πu are related as follows:
ωπ = ωπu | · |−nw/2 ,
(1.2.5)
which is easily checked by looking at the situation at the unramified primes, which
suffices.
For any irreducible, automorphic representation π of G L(n, A F ), there is an
associated L-function L(s, π) = L(s, π∞ )L(s, π f ), called the standard L-function
([J]) of π. It has an Euler product expansion
L(s, πv ),
L(s, π) =
(1.2.6)
v
convergent in a right-half plane. If v is an archimedean place, then one knows (cf.
[La1]) how to associate a semisimple n-dimensional C-representation σ (πv ) of the
Weil group W Fv , and L(s, πv ) identifies with L(s, σv ). We will normalize this correspondence πv → σ (πv ) in such a way that it respects algebraicity. Moreover, if v is
a finite place where πv is unramified, there is a corresponding semisimple conjugacy
class Av (π) (or A(πv )) in GL(n, C) such that
L(s, πv ) = det(1 − Av (π)T )−1 |T =qv−s .
(1.2.7)
We may find a diagonal representative diag(α1,v (π), . . . , αn,v (π)) for Av (π), which
is unique up to permutation of the diagonal entries. Let [α1,v (π), . . . , αn,v (π)] deab
note the unordered n-tuple of complex numbers representing Av (π). Since W F,v
∗
Fv , Av (π) clearly defines an abelian n-dimensional representation σ (πv ) of W F,v .
If 1 denotes the trivial representation of GL(1, A F ), which is cuspidal, we have
L(s, 1) = ζ F (s),
the Dedekind zeta function of F. (Strictly speaking, we should take L(s, 1 f ) on
the left, since the right-hand side is missing the archimedean factor, but this is not
serious.)
The fundamental work of Godement and Jacquet, when used in conjunction with
the Rankin–Selberg theory (see 1.3 below), yields the following:
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10
Dinakar Ramakrishnan
Theorem 1.2.8 ([J]). Let n ≥ 1, and π a non-trivial cuspidal automorphic representation of GL(n, A F ). Then L(s, π) is entire. Moreover, for any finite set S of places
of F, the incomplete L-function
L S (s, π) =
L(s, πv )
v∈
/S
is holomorphic and non-zero in (s) > w + 1 if π has weight w. Moreover, there is
a functional equation
L(w + 1 − s, π ∨ ) = ε(s, π)L(s, π)
with
(1.2.9)
ε(s, π) = W (π)Nπ(w+1)/2−s .
Here Nπ denotes the norm of the conductor Nπ of π, and W (π) is the root number
of π.
Of course when w = 0, i.e., when π is unitary, the statement comes to a more
familiar form. When n = 1, a π is simply an idele class character and this result is
due to Hecke.
By the theory of Eisenstein series, there is a sum operation ([La2], [JS1]):
Theorem 1.2.10 ([JS1]). Given any m-tuple of cuspidal automorphic representations π1 , . . . , πm of GL(n 1 , A F ), . . . , GL(n m , A F ) respectively, there exists an irreducible, automorphic representation π1 · · · πm of GL(n, A F ), n = n 1 +· · ·+n m ,
which is unique up to equivalence, such that for any finite set S of places,
m
L S (s,
m
j =1 π j )
=
L S (s, π j ).
(1.2.11)
j =1
m
Call such a (Langlands) sum π
j =1 π j , with each π j cuspidal, an isobaric automorphic, or just isobaric (if the context is clear), representation. Denote by ram(π)
the finite set of finite places where π is ramified, and let N(π) be its conductor.
For every integer n ≥ 1, set
A(n, F) = {π : isobaric representation of GL(n, A F )}/ ,
(1.2.12)
and
A0 (n, F) = {π ∈ A(n, F)| π cuspidal}.
Put A(F) = ∪n≥1 A(n, F) and A0 (F) = ∪n≥1 A0 (n, F).
Remark 1.2.13. One can also define the analogs of A(n, F) for local fields F, where
the “cuspidal” subset A0 (n, F) consists of essentially square-integrable representations of GL(n, F). See [La3] (or [Ra1]) for details.
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1 Irreducibility and Cuspidality
11
Given any polynomial representation
r : GL(n, C) → GL(N, C),
(1.2.14)
one can associate an L-function to the pair (π, r ), for any isobaric automorphic representation π of GL(n, A F ):
L(s, πv ; r ),
L(s, π; r ) =
(1.2.15)
v
in such a way that at any finite place v where π is unramified with residue field Fqv ,
L(s, πv ; r ) = det(1 − Av (π; r )T )−1 |T =qv−s ,
(1.2.16)
Av (π; r ) = r (Av (π)).
(1.2.17)
with
The conjugacy class Av (π; r ) in GL(N, C) is again represented by an unordered
N-tuple of complex numbers.
The Principle of Functoriality predicts the existence of an isobaric automorphic
representation r (π) of GL(N, A F ) such that
L(s, r (π)) = L(s, π; r ).
(1.2.18)
A weaker form of the conjecture, which suffices for questions like those we are
considering, asserts that this identity holds outside a finite set S of places.
This conjecture is known in the following cases of (n, r):
(2, sym2 ): Gelbart–Jacquet ([GJ])
(2, sym3 ): Kim–Shahidi ([KSh1])
(1.2.19)
(2, sym4 ): Kim ([K])
(4,
2 ):
Kim ([K]).
In this paper we will make use of the last instance of functoriality, namely the
exterior square transfer of automorphic forms from GL(4) to GL(6).
1.3 Rankin–Selberg L-functions
The results here are due to the independent and partly complementary, deep works
of Jacquet, Piatetski-Shapiro and Shalika, and of Shahidi. Let π, π be isobaric automorphic representations in A(n, F), A(n , F) respectively. Then there exists an
associated Euler product L(s, π × π ) ([JPSS], [JS1], [Sh1, Sh2], [MW], [CoPS2]),
which converges in { (s) > 1}, and admits a meromorphic continuation to the whole
s-plane and satisfies the functional equation, which is given in the unitary case by
∨
L(s, π × π ) = ε(s, π × π )L(1 − s, π ∨ × π ),
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(1.3.1)
12
with
Dinakar Ramakrishnan
1
ε(s, π × π ) = W (π × π )N(π × π ) 2 −s ,
where the conductor N(π × π ) is a positive integer not divisible by any rational
prime not intersecting the ramification loci of F/Q, π and π , while W (π × π ) is
the root number in C∗ . As in the Galois case, W (π × π )W (π ∨ × π ∨ ) = 1, so that
W (π × π ) = ±1 when π, π are selfdual.
It is easy to deduce from this the functional equation when π, π are not unitary.
If they are cuspidal of weights w, w respectively, the functional equation relates s to
w + w + 1 − s. Moreover, since π ∨ , π ∨ have respective weights −w, −w , π × π ∨
and π × π ∨ still have weight 0.
When v is archimedean or a finite place unramified for π, π ,
L v (s, π × π ) = L(s, σ (πv ) ⊗ σ (πv )).
(1.3.2)
In the archimedean situation, πv → σ (πv ) is the arrow to the representations of
the Weil group W Fv given by [La1]. When v is an unramified finite place, σ (πv ) is
defined in the obvious way as the sum of one dimensional representations defined by
the Langlands class A(πv ).
When n = 1, L(s, π × π ) = L(s, ππ ), and when n = 2 and F = Q, this
function is the usual Rankin–Selberg L-function, extended to arbitrary global fields
by Jacquet.
Theorem 1.3.3 ([JS1], [JPSS]). Let π ∈ A0 (n, F), π ∈ A0 (n , F), and S a finite
set of places. Then L S (s, π ×π ) is entire unless π is of the form π ∨ ⊗|.|w , in which
case it is holomorphic outside s = w, 1 − w, where it has simple poles.
The Principle of Functoriality implies in this situation that given π, π as above,
there exists an isobaric automorphic representation π π of GL(nn , A F ) such that
L(s, π
π ) = L(s, π × π ).
(1.3.4)
The (conjectural) functorial product is the automorphic analogue of the usual
tensor product of Galois representations. For the importance of this product, see
[Ra1], for example.
One can always construct π π as an admissible representation of GL(nn , A f ),
but the subtlety lies in showing that this product is automorphic.
The automorphy of is known in the following cases, which will be useful to
us:
(n, n ) = (2, 2): Ramakrishnan ([Ra2])
(1.3.5)
(n, n ) = (2, 3): Kim–Shahidi ([KSh1]).
The reader is referred to Section 11 of [Ra4], which contains some refinements,
explanations, refinements and (minor) errata for [Ra2]. It may be worthwhile remarking that Kim and Shahidi use the functorial product on GL(2)×GL(3) which
they construct to prove the symmetric cube lifting for GL(2) mentioned in the previous section (see (1.2.11). A cuspidality criterion for the image under this transfer
is proved in [Ra-W], with an application to the cuspidal cohomology of congruence
subgroups of SL(6, Z).
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1 Irreducibility and Cuspidality
13
1.4 Modularity and the problem at hand
The general Langlands philosophy asserts that if ρ is an n-dimensional -adic representation of GQ , then there is an isobaric automorphic representation π of GL(n, A)
such that for a suitable finite set S of places (including ∞), we have an identity of
the form
(1.4.1)
L S (s, ρ ) = L S (s, π).
When this happens, we will say that ρ is modular, and we will write
ρ ↔ π.
(1.4.2)
One says that ρ is strongly modular if the identity (1.4.1) holds for the full Lfunction, i.e., with S empty.
Special cases of this conjecture were known earlier, the most famous one being
the modularity conjecture for the -adic representations ρ defined by the Galois
action on the -power division points of elliptic curves E over Q, proved recently in
the spectacular works of Wiles, Taylor, Diamond, Conrad and Breuil.
We will not consider any such (extremely) difficult question in this article. Instead we will be interested in the following:
Question 1.4.3. When a modular ρ is irreducible, is the corresponding π cuspidal?
And conversely?
This seemingly reasonable question turns out to be hard to check in dimensions
n > 2.
One thing that is clear is that the π associated to any ρ needs to be algebraic in
the sense of Clozel ([C 1]). To define the notion of algebraicity, first recall that by
Langlands, the archimedean component π∞ is associated to an n-dimensional representation σ∞ , sometimes written σ∞ , of the real Weil group WR , with corresponding
equality of the archimedean L-factors L ∞ (s, ρ ) and L(s, π∞ ). We will normalize
things so that the correspondence is algebraic. One can explicitly describe WR as
C∗ ∪ j C∗ , with j z j −1 = z and j 2 = −1. One gets a canonical exact sequence
1 → C∗ → WR → GR → 1
(1.4.4)
which represents the unique non-trivial extension of GR by C∗ . One has a decomposition
(1.4.5)
σ∞ |C∗ ⊕nj =1 ξ j ,
where each ξ j is a (quasi-)character of C∗ . One says that π is algebraic when every
one of the characters χ j is algebraic, i.e., there are integers p j , q j such that
χ j (z) = z p j z q j .
(1.4.6)
This analogous to having a Hodge structure, which is what one would expect if π
were to be related to a geometric object.
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Dinakar Ramakrishnan
One says that π is regular if for all i = j , χi = χ j . In other words, each character
χ j appears in the restriction of σ∞ (π) to C∗ with multiplicity one. We say (following
[BHR]) that π is semi-regular if the multiplicity of each χ j is at most 2.
When n = 2, any π defined by a classical holomorphic newform f of weight
k ≥ 1 is algebraic and semi-regular. It is regular iff k ≥ 2. One also expects any
Maass waveform ϕ of weight 0 and eigenvalue 1/4 for the hyperbolic Laplacian to
be algebraic; there are interesting examples of this kind coming from the work of
Langlands (resp. Tunnell) on tetrahedral (resp. octahedral) Galois representations ρ
which are even; the odd ones correspond to holomorphic newforms of weight 1. We
will not consider the even situation in this article.
n
2πiz , of weight 2, resp.
Given a holomorphic newform f (z) = ∞
n=1 an q , q = e
k ≥ 3, resp. k = 1, level N and character ω, one knows by Eichler and Shimura, resp.
Deligne ([De]), resp Deligne–Serre ([DeS]), that there is a continuous, irreducible
representation
(1.4.7)
ρ : GQ → GL(2, Q )
such that for all primes p N ,
tr(Fr p |ρ ) = a p
and
det(ρ ) = ωχ k−1 ,
where χ is the -adic cyclotomic character of GQ , given by the Galois action on the
-power roots of unity in Q, and Fr p is the geometric Frobenius at p, which is the
inverse of the arithmetic Frobenius.
1.5 Parity
We will first first introduce this crucial concept over the base field Q, as that is what
is needed in the remainder of the article.
We will need to restrict our attention to those isobaric forms π on GL(n)/Q
which are odd in a suitable sense. It is instructive to first consider the case of a
classical holomorphic newform f of weight k ≥ 1 and character ω relative to the
congruence subgroup 0 (N). Since 0 (N) contains −I , it follows that ω(−1) =
(−1)k . One could be tempted to call a π defined by such an f to be even (or odd)
according as ω is even (or odd), but it would be a wrong move. One should look
not just at ω, but at the determinant of the associated ρ , i.e., the -adic character
ωχ k−1 , which is odd for all k ! So all such π defined by holomorphic newforms are
arithmetically odd. The only even ones for GL(2) are (analytic Tate twists of) Maass
forms of weight 0 and Laplacian eigenvalue 1/4.
The maximal abelian quotient of WR is R∗ , and the restriction of the abelianization map to C∗ identifies with the norm map z → |z|. So every (quasi)-character
ξ of WR identifies with one of R∗ , given by x → sgn(x)a |x|t for some t, with
a ∈ {0, 1}. Clearly, ξ determines, and is determined by (t, a). If π is an isobaric automorphic representation, let σ∞ [ξ ] denote, for each such ξ , the ξ -isotypic component of σ∞ . The sign group R∗ /R∗+ acts on each isotypic component. Let m + (π, ξ )
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1 Irreducibility and Cuspidality
15
(resp. m − (π, ξ )) denote the multiplicity of the eigenvalue +1 (resp. −1), under the
action of R∗ /R∗+ on σ∞ (ξ ).
Definition 1.5.1. Call an isobaric automorphic representation π of GL(n, A) odd if
for every one-dimensional representation ξ of WR occurring in σ∞ ,
|m + (π, ξ ) − m − (π, ξ )| ≤ 1.
Clearly, when the dimension of σ∞ [ξ ] is even, the multiplicity of +1 as an eigenvalue of the sign group needs to be equal to the multiplicity of −1 as an eigenvalue.
Under this definition, all forms on GL(1)/Q are odd. So are the π on GL(2)/Q
which are defined by holomorphic newforms of weight k ≥ 2. The reason is that
π∞ is (for k ≥ 2) a discrete series representation, and the corresponding σ∞ (π) is
an irreducible 2-dimensional representation of WR induced by the (quasi)-character
z → z −(k−1) of the subgroup C∗ of index 2, and our condition is vacuous. On the
other hand, if k = 1, σ∞ (π) is a reducible 2-dimensional representation, given by
1 ⊕ sgn. The eigenvalues are 1 on σ∞ (1) and −1 on σ∞ (sgn). On the other hand,
a Maass form of weight 0 and λ = 1/4, the eigenvalue 1 (or −1) occurs with multiplicity 2, making the π it defines an even representation. So our definition is a good
one and gives what we know for n = 2.
For any n, note that if π is algebraic and regular, it is automatically odd. If π is
algebraic and semi-regular, each isotypic space is one or two-dimensional, and in the
latter case, we want both eigenvalues to occur for π to be odd.
Finally, if F is any number field with a real place u, we can define, in exactly the
same way, when an algebraic, isobaric automorphic representation of GL(n, A F ) is
arithmetically odd at u. If F is totally real, then we say that π is totally odd if it is
so at every archimedean place.
2 The first step in the proof
Let ρ , π be as in Theorem A. Since ρ is irreducible, it is in particular semisimple.
Suppose π is not cuspidal. We will obtain a contradiction.
Proposition 2.1. Let ρ , π be associated, with π algebraic, semi-regular and odd.
Suppose we have, for some r > 1, an isobaric sum decomposition
π
r
j =1 η j ,
(2.2)
where each η j is a cuspidal automorphic representation of GL(n j , A), with n j ≤ 2
(∀ j ). Then ρ cannot be irreducible.
Corollary 2.3. Theorem A holds when π admits an isobaric sum decomposition such
as (2.2) with each n j ≤ 2. In particular, it holds for n ≤ 3.
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16
Dinakar Ramakrishnan
Proof of Proposition. The hypothesis that π is algebraic and semi-regular implies
easily that each η j is also algebraic and semi-regular. Let Jm denote the set of j
where n j = m.
First look at any j in J1 . Then the corresponding η j is an idele class character.
Its algebraicity implies that, in classical terms, it corresponds to an algebraic Hecke
character ν j . By Serre ([Se]), we may attach an abelian -adic representation ν j, of
GQ of dimension 1. It follows that for some finite set S of places containing ,
L S (s, ν j, ) = L S (s, η j )
whenever n j = 1.
(2.4)
Next consider any j in J2 . If σ (η j,∞ ) is irreducible, then a twist of η j must
correspond to a classical holomorphic newform f of weight k ≥ 2. Moreover, the
algebraicity of η j forces this twist to be algebraic. Hence by Deligne, there is a
continuous representation
τ j, : GQ → GL(2, Q ),
ramified only at a finite of primes such that at every p =
is unramified,
tr(Fr p |τ j, ) = a p (η j ),
(2.5)
where the representation
(2.6)
and the determinant of τ j, corresponds to the central character ω j of η j . Moreover,
τ j, is irreducible, which is not crucial to us here.
We also need to consider the situation, for any fixed j ∈ J2 , when σ (η j,∞ ) is
reducible, say of the form χ1 ⊕ χ2 . Since η j is cuspidal, by the archimedean purity
result of Clozel ([C 1]), χ1 χ2−1 must be 1 or sgn. The former cannot happen due to
the oddness of π. It follows that η j is defined by a classical holomorphic newform f
of weight 1, and by a result of Deligne and Serre ([DeS]), there is a 2-dimensional adic representation τ j, of GQ with finite image, which is irreducible, such that (2.6)
holds.
Since the set of Frobenius classes Fr p , as p runs over primes outside S, is dense
in the Galois group by Tchebotarev, we must have, by putting all these cases together,
ρ
(⊕ j ∈ J1 ν j, ) ⊕ (⊕ j ∈ J2 τ j, ),
(2.7)
which contradicts the irreducibility of ρ , since by hypothesis, r = |J1 | + |J2 | ≥ 2.
3 The second step in the proof
Let ρ , π be as in Theorem A. Suppose π is not cuspidal. In view of Proposition 2.1,
we need only consider the situation where π is an isobaric sum j η j , with an η j
being a cusp form on GL(m)/Q for some m ≥ 3.
Proposition 3.1. Let ρ , π be associated, with π an algebraic cusp form on GL(n)/Q
which is semi-regular and odd. Suppose we have an isobaric sum decomposition
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1 Irreducibility and Cuspidality
π
η
η,
17
(3.2)
where η is a cusp form on GL(3)/Q and η is an isobaric automorphic representation
of GL(r, A) for some r ≥ 1. Moreover, assume that there is an r -dimensional -adic
representation τ of GQ associated to η . Then we have the isomorphism of GQ modules:
∨
2
(ρ ) ⊕ τ ⊕ sym2 (τ ).
(3.3)
ρ ∨ ⊕ (ρ ⊗ τ )
Corollary 3.4. Let ρ , π be associated, with π algebraic, semi-regular and odd. Suppose π admits an isobaric sum decomposition such as (3.2) with r ≤ 2. Then ρ is
reducible.
Proposition 3.1 ⇒ Corollary 3.4. When r ≤ 2, η is either an isobaric sum of
algebraic Hecke characters or cuspidal, in which case, thanks to the oddness, it is
defined by a classical cusp form on GL(r )/Q of weight ≥ 1. In either case we have,
as seen in the previous section, the existence of the associated -adic representation
τ , which is irreducible exactly when η is cuspidal. Then by the proposition, the
decomposition (3.3) holds. If r = 1 or r = 2 with η Eisensteinian, (3.3) implies
that a 1-dimensional representation (occurring in τ ) is a summand of a twist of either
ρ or ρ ∨ . Hence the corollary.
Combining Corollary 2.3 and Corollary 3.4, we see that the irreducibility of ρ
forces the corresponding π to be cuspidal when n ≤ 4 under the hypotheses of Theorem A. So we obtain the following:
Corollary 3.5. Theorem A holds for n ≤ 4.
Proof of Proposition 3.1. By hypothesis, we have a decomposition as in (3.2), and
an -adic representation τ associated to η .
As a short digression let us note that if η were essentially selfdual and regular,
we could exploit its algebraicity, and by appealing to [Pic] associate a 3-dimensional
-adic representation to η. The Proposition 3.1 will follow in that case, as in the proof
of Proposition 2.1. However, we cannot (and do not wish to) assume either that η is
essentially selfdual or that it is regular. We have to appeal to another idea, and here
it is.
Let S be a finite set of primes including the archimedean and ramified ones. At
any p outside S, let π p be represented by an unordered (3 + r )-tuple {α1 , . . . , α3+r }
of complex numbers, and we may assume that η p (resp. η p ) is represented by
{α1 , α2 , α3 } (resp. {α4 , . . . , α3+r }. It is then straightforward to check that
L(s, π p ;
2
) = L(s, η∨p )L(s, η p × η p )L(s, η ;
2
).
(3.6)
One can also deduce this as follows. Let σ (β) denote, for any irreducible admissible
representation β of GL(m, Q p ), the m-dimensional representation of the extended
Weil group WQ p = WQ p × SL(2, C) defined by the local Langlands correspondence
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